Biomedical Engineering Reference
In-Depth Information
k
k
off
on
Note that these two roots
have the dimension of a concentration and that
is the
equilibrium constant. In order to simplify the notations, we note
2
k
k
æ
S
ö
off
off
c
ˆ
=
c
-
-
G
+
4
c
ç
÷
0
0
0
è
k
V
ø
k
on
on
c
has also the dimension of a concentration. Using the notations
c
,
c
+
, and
c
-
, it is
possible to show that the solution of (7.67) is
1
-
c
=
+
c
1
1
(1
(7.70)
-
k c t
ˆ
-
k c t
ˆ
e
+
-
e
)
on
on
-
c
ˆ
c
-
c
0
The kinetics of surface concentration is obtained by substituting the concentration
kinetics from (7.70) in (7.65)
V
c
G =
(
-
c
)
(7.71)
0
S
At first glance (7.67) seems to be somewhat complicated, but this is not really the
case. Let us examine three different cases:
1. First, it is easy to see that if
V
®
¥, then S/
V
®
0 and we obtain
c = c
0
, which
is the expected result for a semi-infinite case.
2. Second, suppose that
k
off
=
0
and that the number of targets is larger than the
number of ligands (initial hybridization sites).
n
c V
targets
ligands
0
0
N
=
=
>
1
n
G
S
The functionalized surface will end being saturated by the immobilized targets. One
finds first that
c
-
=
0
and
=
S
c
ˆ
c
-
G
. Then, (7.62) collapses to
0
0
V
1
c
=
S
æ
S
ö
æ
ö
æ
ö
-
k
c
-
G
t
-
k
c
- G
t
1
1
ç
÷
ç
÷
on
0
0
on
0
0
è
ø
è
ø
V
ç
V
÷
e
+
1
-
e
c
æ
S
ö
ç
÷
0
c
-
G
è
ø
ç
÷
0
0
è
ø
V
S
c c
V
and G ® G
0
3. Third, that
k
off
=
0
and that the number of targets is smaller than the number
of ligands (initial hybridization sites)
By letting
t
®
¥ in the preceding equation,
® - G
0
0
n
c V
targets
ligands
0
0
N
=
=
<
1
n
G
S
S
-
=
The functionalized surface cannot be saturated. It is easy to see that
c
c
-
G
0
0
V
æ
S
ö
and
c
ˆ
= -
c
-
G
ç
÷
0
0
è
ø
V
